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%{\LARGE\bf 上海立信会计金融学院期终考试卷 --- 试题纸} \hspace{0.3cm} {\Large \underline{ A }卷 }
{\Large\bf \H 上海立信会计金融学院期终考试卷 } \hspace{0.3cm} {\Large \underline{ A }卷 }

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{\large \bf \H 2023 $\sim$ 2024 学年 第 二 学期 }

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{\large \bf \H \underline{ \emph{2022级跨学科跨专业选修课班级} } 《\underline{ \emph{复变函数} }》 课程代码：\underline{ 162250220 }  }

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{\H（本场考试属\underline{  开  }卷考试，考试时间共\underline{  90  }分钟，不准使用计算器）共\underline{  4  }页 }
%{\large （本场考试属闭卷考试，考试时间 90 分钟，禁止使用计算器） }

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%{\large 本考试卷共 4 页，请在本考试卷上答题。}

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班级 \underline{\hspace{3.5cm}} 学号 \underline{\hspace{3.5cm}} 姓名 \underline{\hspace{3.5cm}} 

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{\H
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题号 &一&二&三&四&五&六&七&八&九&十&总分&合成人签名&审核人签名 \\
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%应得分&15&15&15&15&15&15&10&100 \\
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得分 $\,\,\,\,\,\,\,\,$ &&&&&&&&&&&&& \\
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本次考试共10题，每题10分。
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\begin{enumerate}

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\item %1
函数 $w=z^2$ 将 $z$ 平面上的直线 $x-y-1=0$ 变成 $w$ 平面上的什么曲线？

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\item %2
验证函数 $f(z) = 3x^2-3y^2-4x+6xyi-4yi+6$ 在复平面上满足柯西黎曼方程，证明 $f(z)$ 在复平面上解析，并求其导数。

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\item %3
考虑初等多值函数 $f(z)=3^z$. 
计算 $f(1+2i)$ 的所有值。

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\item %4
设解析函数 $f(z)=\sqrt[3]{z}$ 定义在去掉负实轴的复平面上，并且 $f(1)=-\frac{1}{2}+\frac{\sqrt{3}}{2}i$, 试求 $f(1-i)$ 的值。

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\item %5
设积分路径 $C$ 是单位圆周的一部分，逆时针连接点 $1$ 和点 $-1$. 使用参数方程法，计算积分 $$\int_C (y+ix^2)dz. $$

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\item %6
设 $C$ 为圆周 $| z-3|=2$. 使用柯西积分公式，计算 $$\int_C \frac{dz}{z^2(z^2-4)}.$$

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\item %7
使用柯西-阿达马公式计算幂级数 $\sum\limits_{n=0}^{\infty} \cos(n)z^n$ 的收敛半径。

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\item %8
设有泰勒展开 $ \frac{1}{z^2+2z+3} = \sum\limits_{n=0}^{\infty} c_nz^n$. 
求出该幂级数的收敛范围与系数之间的递推关系式。 

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\item %9
设复平面上有两个同向相似的三角形，顶点分别为 $z_1,z_2,z_3$ 和 $w_1,w_2,w_3$. 证明
\begin{equation*}
\begin{vmatrix}
z_1 & w_1 & 1 \\ 
z_2 & w_2 & 1 \\ 
z_3 & w_3 & 1 \\ 
\end{vmatrix}.
\end{equation*}


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\item %10
设 $C$ 为圆周 $|z|=2$, 计算积分 
$$
\int_C \frac{\sin(z)}{(z-i)^4}dz. 
$$


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\end{enumerate}
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